Proton Spin Structure From Monte Carlo Global Qcd Analyses

Transcription

1 College of William and Mary W&M ScholarWorks Dissertations, Theses, and Masters Projects Theses, Dissertations, & Master Projects Summer 8 Proton Spin Structure From Monte Carlo Global Qcd Analyses Jacob Ethier College of William and Mary - Arts & Sciences, jethier@ .wm.edu Follow this and additional works at: Part of the Physics Commons Recommended Citation Ethier, Jacob, "Proton Spin Structure From Monte Carlo Global Qcd Analyses" (8). Dissertations, Theses, and Masters Projects. Paper This Dissertation is brought to you for free and open access by the Theses, Dissertations, & Master Projects at W&M ScholarWorks. It has been accepted for inclusion in Dissertations, Theses, and Masters Projects by an authorized administrator of W&M ScholarWorks. For more information, please contact scholarworks@wm.edu.

2 Proton Spin Structure from Monte Carlo Global QCD Analyses Jacob Ethier Deltona, Florida Master of Science, College of William & Mary, 5 Bachelor of Arts, Stetson University, 3 ADissertationpresentedtotheGraduateFaculty of The College of William & Mary in Candidacy for the Degree of Doctor of Philosophy Department of Physics College of William & Mary May 8

3 c 8 Jacob Ethier All rights reserved.

4

5 ABSTRACT Although significant progress has been made in recent years in understanding the composition of the proton s spin from its quark and gluon constituents, a complete picture has yet to emerge. Such information is encoded in spin-dependent parton distribution functions (PDFs) that, as a consequence of being inherently nonperturbative, must be extracted through global QCD analyses of polarized lepton-nucleon and proton-proton collisions. Experiments that measure a final state hadron from these reactions are particularly useful for separating the individual quark and anti-quark polarizations, but require knowledge of parton-to-hadron fragmentation functions (FFs) to describe theoretically. In this thesis, we present a new approach to global QCD analyses, that were performed recently by the Je erson Lab Angular Momentum (JAM) Collaboration to determine the spin PDFs and FFs from deep inelastic scattering (DIS), semi-inclusive DIS, and single inclusive electron-positron annihilation observables. While previous global QCD studies typically used a single minimization procedure, the JAM Collaboration applies a robust Monte Carlo fitting methodology to extract the central values and uncertainties of the relevant distributions. The results from these JAM global QCD analyses, which include a first ever simultaneous fit of the spin PDFs and FFs, resolve a long-standing puzzle regarding the strange quark polarization and provide new information about the proton spin structure.

6 TABLE OF CONTENTS Acknowledgments Dedication List of Tables iv v vi List of Figures vii CHAPTER Introduction Perturbative QCD and Factorization History of Proton Spin Structure Outline High Energy Scattering Observables Deep inelastic scattering Semi-inclusive deep inelastic scattering Single-inclusive e + e annihilation Observables in Mellin moment space Corrections Beyond the Parton Model Higher Twist and Target Mass Corrections in DIS Mellin moments and spin sum rules Nuclear Corrections Application to unpolarized electron-deuteron scattering Application to polarized electron- 3 He scattering i

7 4 Aspects of Fitting Bayesian Approach Method of Maximum Likelihood Hessian Error Propogation Monte Carlo Techniques Data Resampling and Cross Validation Iterative Monte Carlo (IMC) Nested Sampling Parameterizations Helicity distributions and higher twist Fragmentation functions Penalties and prior information Experimental Data Deep inelastic scattering observables Single-inclusive e + e annihilation observables Semi-inclusive deep-inelastic observables Results from JAM Global QCD Analyses Iterative Monte Carlo Analysis of Spin PDFs Impact of Je erson Lab data JAM5 Distributions and Moments Iterative Monte Carlo Analysis of FFs JAM6 Fragmentation Functions Simultaneous Extraction of PDFs and FFs JAM7 Spin PDFs and Fragmentation Functions Resolution of the Strange Polarization Discrepancy ii

8 5.3.3 Lowest Moments of Spin PDF Combinations Conclusion and Outlook Summary of Results Future of PDF and FF Extraction APPENDIX A Hard Scattering Coe cients in Mellin Space APPENDIX B Splitting Functions in Mellin Space Bibliography iii

9 ACKNOWLEDGMENTS First and foremost, I want to thank my advisor Wally Melnitchouk, who introduced me to QCD phenomenology while I was a young and naive undergraduate summer student at Je erson Lab. I will always be grateful to Wally for that opportunity, and for the years since, during which he has proved to be an indispensable mentor and teacher. I would also like to thank Nobuo Sato, for his invaluable contributions to the JAM Collaboration, and for his helpful guidance and advice during my time as a graduate student. I will always appreciate the many discussions we ve had on fitting techniques and perturbative QCD formalism. Thanks to all of the other members of the JAM Collaboration who have helped make this work possible, and to the members of my defense committee for their feedback and suggestions. Thanks to my friends and family, especially my parents, for their support during my studies. Finally, I am very grateful to my wife, Lindsay, and our two children, for their unwavering love, support, and patience while I worked and traveled. They have been a constant source of encouragement and comfort, particularly during di cult times. Our cat, Gypsy, has kept me company while writing in the nighttime hours, so perhaps I should thank her as well. iv

10 To my wife, Lindsay, and our two children Emerson and Colin. v

11 LIST OF TABLES 3. E ective polarization parameters Impact of hadron final state mass squared cuts on global fits Impact of four-momentum transfer squared cuts on global fits Summary of the JAM global QCD analyses Summary of values for inclusive DIS data sets Lowest moments of leading and higher twist functions d moments of the proton and neutron g, structure functions Summary of values for single-inclusive e + e annihilation data sets Summary of values for semi-inclusive DIS data sets vi

12 LIST OF FIGURES. Feynman diagram of deep inelastic scattering Feynman diagram of semi-inclusive deep inelastic scattering Feynman diagram of single-inclusive e + e -annihilation Quasielastic electron deuteron scattering cross section Quasielastic electron deuteron scattering cross section Quasielastic electron deuteron scattering cross section Spin-dependent xg and xg structure functions of the neutron and 3 He Polarization asymmetries A and A of the neutron and 3 He d moment of the neutron and 3 He Workflow of the iterative Monte Carlo fitting strategy Convergence of the Convergence of the fragmentation function prior volume Kinematic coverage in x and Q of the polarized inclusive DIS data sets used in the JAM analyses Dependence on W cut of moments of leading and higher twist distributions Dependence on Q cut of moments of leading and higher twist distributions Proton longitudinal polarization asymmetries A p k and Ap from non-je erson Lab experiments vii

13 5. Proton transverse polarization asymmetries A p? and Ap from non-je erson Lab experiments Proton longitudinal polarization asymmetries A p k 5.4 Proton longitudinal polarization asymmetries A p k from Je erson Lab from Je erson Lab Deuteron longitudinal polarization asymmetries A d k and Ad from non-je erson Lab experiments Deuteron transverse polarization asymmetries A p? from non-je erson Lab experiments Deuteron longitudinal polarization asymmetries A d k 5.8 Deuteron longitudinal polarization asymmetries A d k from Je erson Lab.. 9 from Je erson Lab He longitudinal (A He k, AHe ) and transverse (A He?, AHe )polarizationasymmetries Impact of Je erson Lab data on leading and higher twist distributions Final JAM5 results for the leading and higher twist distributions Comparison of JAM leading twist distributions with other parameterizations d moments of the proton and neutron Normalized yield of IMC fits versus /N dat Ratio of experimental single-inclusive e + e cross sections to the fitted values versus z for pion production Ratio of experimental single-inclusive e + e cross sections to the fitted values versus z for kaon production Final results for the extracted fragmentation functions Iterative convergence of the + and K + fragmentation functions Evolution of the + and K + fragmentation functions Comparison of the JAM fragmentation functions with other parameterizations viii

14 5. Comparison of the JAM fragmentation functions with other parameterizations at the scale Q = M Z Proton and deuteron longitudinal polarization asymmetries A h± for charged pion and kaon production Final JAM7 leading twist helicity distributions Semi-inclusive polarization asymmetries A p and A K d Final JAM7 fragmentation functions Lowest moments of the spin-dependent PDFs ix

15 PROTON SPIN STRUCTURE FROM MONTE CARLO GLOBAL QCD ANALYSES

16 CHAPTER Introduction The discovery of the nucleus by Rutherford in the early th century was a crucial turning point in our understanding of atomic structure. The collection of positively charged protons in the nucleus contradicted what was understood about the electromagnetic force: that particles of like charge repel one another. Clearly there existed an additional force, one that acted more strongly over smaller distance scales, that bound the protons and neutrons in the nucleus together. The following decades of particle scattering experiments revealed that these protons and neutrons, or nucleons, were not fundamental particles, but instead constructed of point-like constituents known as quarks. A picture of the nucleon as a dynamical system of these quarks, which carry a quantum property called color, and their interactions via gluons, the particle mediating the strong force, quickly emerged as the leading explanation for the formation of nucleons and nuclei. Thus, quantum chromodynamics (QCD) was born as the theory that describes the interactions of quarks and gluons, the constituents of all atomic nuclei.

17 3. Perturbative QCD and Factorization Before discussing further about the structure of the nucleon, it is important to understand two defining features of QCD, namely confinement and asymptotic freedom. Both can be understood by considering the strength of the quark-gluon interactions g s,ormore commonly the QCD strong coupling s = gs/(4 ), a function that depends on the number of active quark flavors N f and a renormalization scale µ R that arises from regulating ultraviolet divergences. In energy regions where the number of quark flavors is constant, the approximate analytical solution for s at lowest order is s (µ R ) ' (C A 4N f T R )ln(µ R / ) (.) where C A and T R are SU(3) color factors []. The additional scale is a QCD constant of integration, and indicates the region in which s becomes divergent (µ R ). By setting the renormalization scale to the momentum transfer of a particle scattering process, µ R = Q,itbecomesclearthattherearetwodistinctregionsforthestrengthofthecoupling. At su ciently high energies ( Q), or very small distance scales, the coupling becomes small and the quarks can be treated essentially as free particles in the nucleon. This is known as asymptotic freedom, and is the basis of perturbative QCD (pqcd). Within the pqcd framework, experimental observables can be formulated as a series expansion in s.suchpredictivepowerislost,however,ifthescaleofthereactionbecomes small (Q ). This is the region of confinement, and, as the name suggests, is the reason why free quarks and gluons have not been observed directly. The nonperturbative region of QCD has yet to be fully understood, especially the process in which the quarks and gluons become hadrons, and remains an important challenge for QCD studies such as this work.

18 4 The structure of the nucleon in terms of its constituent quarks and gluons, or more generally partons, can be determined from high energy particle collisions where the reactions can be described from pqcd. However, even in pqcd calculations of observables, one finds large logarithms that spoil the convergence of the perturbative series (i.e. logarithms that depend on a scale that is of the order in the definition of the strong coupling). This is supported by the fact that experiments can only measure bound state hadrons, and therefore observables must have dependence on the long-distance energy scales related to the hadronization process. This issue is resolved by factorizing the perturbative and nonperturbative regions in calculations of high energy scattering observables []. Consider, for example, deep inelastic scattering (DIS) where a lepton scatters from a proton target. In the limit of large momentum transfer Q, theleadingcontributiontothecrosssectioncanbeapproximated as a convolution of the hard scattering cross section, dˆf, withasoftnonperturbative function, f, d (x, Q ) ' X f Z x d x f,q dˆf(,q ), (.) where higher order /Q corrections can be safely neglected. At leading order (LO) in s, the cross section has a probabilistic interpretation in that dˆf is the probability for an electron to scatter from a quark of flavor f and f(x) isthepartondistributionfunction (PDF) that describes the probability for the struck quark to carry momentum fraction x of the total proton momentum. The di erential cross section for the electron-proton scattering as a function of x and Q is then the sum over all possible quark flavors in the proton. Factorization is a powerful tool that allows us to determine the partonic momentum structure of hadrons through PDFs. Because these objects are nonperturbative, they

19 5 cannot be directly computed in pqcd but are determined in analyses of high energy experimental data instead. Significant progress has been made over the past several decades to constrain PDFs through global QCD analyses [3 6]. In addition to momentum distributions, experiments which polarize the scattering particles can provide information about the spin structure of the proton as well. This is encoded in spin-dependent PDFs, the determination of which will be the focus of this thesis.. History of Proton Spin Structure The first measurement of the proton spin structure from polarized DIS by the European Muon Collaboration (EMC) revealed a rather surprising result [7]. While one might expect the quarks to carry all of the nucleon spin from a naïve quark model picture, a more careful prediction from Ellis and Ja e [8] suggested the quark contribution to the proton spin at the EMC scale, assuming zero strange quark polarization, is (Q EMC ).6, roughly 6% of the total spin [9]. However, the EMC analysis determined (Q EMC )., a factor of six di erence! The discovery that the quarks carry such a small fraction of the proton spin became famously known as the proton spin crisis. Consequently, various attempts to explain the proton spin puzzle quickly emerged. One such e ort proposed a large cancellation of the quark contribution through a gluonic term in generated by the axial anomaly []. This prompted intense interest in measuring the gluon polarization in the proton, G. While subsequent DIS experiments struggled to find any indication of a nonzero gluon spin contribution, recent measurements from proton-proton collisions at the Relativistic Heavy Ion Collider (RHIC) provided evidence for a small G [, ]. Unfortunately, the extracted value from RHIC data was significantly smaller than what was needed to account for the EMC result [3]. Subsequent experiments also failed to validate explanations that suggested a large negative sea

20 6 polarization in the proton. The unaccounted spin is yet to be identified, however, the attention is now focused on orbital angular motion of the quarks and gluons [9, 4]. Such information can be accessed from transverse momentum dependent (TMD) distributions or generalized parton distributions (GPDs) which are functions of quark transverse momentum k T and transverse spatial distance b, respectively,inadditiontothequarklongitudinalmomentumfraction. The focus of this work, however, is on the collinear helicity distributions, f = f " f #, defined here as the di erence between parton distributions with spin aligned (") andantialigned (#) with the proton spin. Much of the information about quark spin PDFs has come from global analyses of polarized DIS experiments. Tremendous improvement in both experimental and theoretical precision has led to well determined distributions for the valence quarks [5 ]. In more recent years, additional information has come from polarized semi-inclusive deep inelastic scattering (SIDIS) and single spin asymmetries in W ± boson production, which are more sensitive to the sea quark polarizations. The total strange and anti-strange helicity distribution s + = s+ s, inparticular, has been of interest in the last decade. Since polarized DIS observables are weakly dependent on the strange polarization, global analyses typically use SU(3) flavor symmetric (SU(3) f ) constraints from weak baryon decays to extract such information. However, this requires the integral of s + over all parton momentum fraction x to have a value of. at Q GeV,anoverallnegativecontributiontothespinoftheproton.Furthermore, fit parameters that control the shape of the strange polarization in the region of large x are often fixed such that s + (x) containsanegativepeakatx.. While an entirely negative s + has been determined from inclusive DIS data, global fits that incorporated polarized SIDIS measurements showed a dramatic shift in the shape of the strange PDF, changing sign and becoming positive in the intermediate-x region instead [, ]. Various questions soon emerged about the cause of this discrepancy, which

21 7 became known as the strange polarization puzzle []. If polarized SIDIS observables prefer a positive strange helicity at x., why does this not appear as a possible distribution obtained in analyses of DIS only? Is there a tension between DIS and SIDIS data? Such questions can be answered through rigorous Monte Carlo based QCD analyses. By analyzing experimental data with robust statistical procedures, we can obtain a better understanding of the spin structure of the proton..3 Outline Much of the work presented here has already been published and can be found in Refs. [5, 5]. We will begin with a review of the theoretical formalism for polarized DIS, SIDIS, and unpolarized single-inclusive e + e annihilation (SIA) processes in Chapter. The pqcd expressions for DIS and SIDIS are not only written as a series expansion in the strong coupling s,butalsocontainhigherorder/q corrections from higher twist operators in the operator product expansion (OPE). Furthermore, experiments often perform DIS and SIDIS with nuclear targets such as deuterium and 3 He, which require careful treatment to extract quark spin structure information [, 3]. Both higher twist and nuclear corrections are discussed in detail in Chapter 3. An essential part of any global QCD analysis is the methodology one uses to fit the experimental data. We will discuss various aspects of fitting, including both Hessian and Monte Carlo based fitting techniques, in Chapter 4. Function parameterizations and features of experimental data included in the global fits will also be discussed in this chapter. Following details on PDF extraction methods, we will present results from three di erent global analyses in Chapter 5. The first studies the impact of Je erson Lab DIS data and higher twist corrections on the parton spin dependent distributions [5]. The second is the first Monte Carlo analysis of e + e annihilation to extract fragmentation

22 8 functions (FFs) [4], which play a significant role in spin PDF extractions from SIDIS observables and will be introduced in Chapter. Lastly, we will present results from an analysis that, for the first time, fit simultaneously the quark helicity distributions and FFs in a combined QCD analysis of DIS, SIDIS, and SIA experimental data [5]. The analysis emphasizes in particular the impact of SIDIS data on the sea quark polarizations and resolves the long-standing puzzle regarding the strange helicity shape. As a result of the three subsequent global fits presented in this work, we obtain new and reliable information about the proton spin structure. The results will be summarized in Chapter 6, followed by a discussion about the future of PDF and FF extraction.

23 CHAPTER High Energy Scattering Observables Collinear factorization is the theoretical foundation for constructing observables in global QCD analyses of high energy scattering data. In this chapter, we review the formalism to describe DIS, SIDIS, and SIA measurements within this framework. In addition, we give the general expressions for the observables in Mellin moment space and discuss the benefits of implementing the Mellin technique in global QCD fits.. Deep inelastic scattering In polarized inclusive DIS, a lepton with spin aligned (") oranti-aligned(#) withits direction of motion scatters from a polarized nucleon target of mass M via the exchange of a virtual photon (see Fig..). The di erential cross section, neglecting the lepton mass, can be expressed theoretically as a contraction of the leptonic tensor L µ and hadronic tensor W µ, d dxdy = y Q 4 L µ W µ, (.) 9

24 p X FIG..: Feynman diagram of deep inelastic scattering. The incoming lepton ` scatters inelastically from the nucleon target p via the exchange of a virtual photon. The target remnants fragment into final state hadrons X, but only the outgoing lepton ` is measured in the final state. where Q = q is the squared four-momentum transfer, y = /E =(E E )/E is the lepton fractional energy loss, and = e /4 is the electromagnetic fine structure constant. In Eq. (.), the dependence on the Bjorken scaling variable x = Q /M is implicit in the hadronic tensor W µ. The lepton-photon vertex in the upper part of Fig.. is described by the leptonic tensor, which can be computed exactly in quantum electrodynamics (QED) as L µ =(`µ` + `µ` ` `g µ i µ ` ` ), (.) for an incoming and outgoing lepton with four-momenta `µ and `µ, respectively. Here µ is the antisymmetric Levi-Civita tensor and = ± representsthehelicityofthe incoming lepton. On the other hand, the hadronic tensor, which describes the photon-nucleon interaction, is di cult to compute from first principles due to the nonperturbative nature of the bound nucleon. However, respecting current conservation and parity, we can write the

25 most general form for W µ as a linear combination of scalar coe cients F, and g,, W µ = + g µ + qµ q F q (x, Q ) P µ q µ P q P q + i µ q S P q g (x, Q )+i µ q P q F (x, Q ) q P q q P q S P S q P q g (x, Q ), (.3) where P µ and q µ are the four-momenta of the nucleon and photon, respectively, and S is the nucleon spin four-vector, with S = M and S P =. Thescalarcoe cients F (x, Q ), F (x, Q ), g (x, Q ), and g (x, Q ) are known as structure functions, and will be discussed in the context of pqcd later in this section. Experimental quantities typically measured in polarized DIS are cross section asymmetries, where the di erence between di erent lepton and target spin configurations is observed with respect to the spin averaged cross section. The most general spin asymmetry can be defined in terms of spherical polar angles and,describingthedirectionof the target polarization relative to the virtual photon momentum vector q, A = # " # + " = cos p A +sin cos p ( )A + R, (.4) where "# d "# /dxdy corresponds to Eq. (.) for a lepton with helicity (") =+or (#) =. The general spin asymmetry depends on the ratio R of the longitudinal to transverse photon absorption cross sections, which will be defined in terms of the structure functions later in this section. In addition, Eq. (.4) depends on the kinematic variable, which is given by = ( y) y +( y) + y, (.5)

26 with =4M x /Q. By polarizing the target parallel (*) orperpendicular()) tothe beam direction, Eq. (.4) can be separated into longitudinal and transverse spin asymmetries, A k = A? = #* "* #* + "* = D(A + A ), (.6) #) ") #) + ") = d(a A ), (.7) where the photon polarization factors D, d,, and are defined as y( y)( + y) D = ( + )y +(4( y) y )( + R), p 4( y) y d = D, (.8) y = 4( y) y ( y)( + y), = y + y. The virtual photoproduction asymmetries A and A that appear in Eqs. (.4), (.6), and (.7) can be decomposed into simple ratios of the spin-dependent to spin-averaged structure functions, A = (g g ), A = (g + g ). (.9) F F Extracting these quantities from polarized electron-nucleon scattering requires information about the ratio R, whichcanalsobewrittenintermsofthestructurefunctions, R = ( + )F xf xf. (.) The most fundamental observables that can be measured in DIS, then, are the polarized g i and unpolarized F i (i =, ) structure functions.

27 Both the spin-averaged and spin-dependent structure functions can be described by a series expansion in powers of /Q, the origin of which will be introduced in Chapter 3. For now, the discussion will focus only on the leading contribution to the structure functions, which are well defined in pqcd. The leading term in g is given by a convolution of the polarized PDFs ( q + = q + q, g) withthehardscatteringcoe cients( C q,g ), g (x, Q )= X q e q ( Cq q + )(x, Q )+ ( C g g)(x, Q ) + O, (.) Q where e q is the quark electric charge and represents the standard convolution integral C f = R (dˆx/ˆx)c(ˆx)f(x/ˆx). In principle, the polarized PDFs and coe x 3 cient functions depend on a renormalization scale µ R that originates from regulating ultraviolet divergences. Typically this is set to the hard scale Q, as was done in Eq. (.), but can be varied to estimate theoretical uncertainty. Note also that the expression for the structure function is true to all orders in pqcd since the expansion in s is implicit in the hard coe cient functions, C i (ˆx, Q )= C () i (ˆx, Q )+ s(q ) 4 C () i (ˆx, Q )+O( s), (.) where µ R = Q was also set for the strong coupling s. The leading contribution to the unpolarized structure functions F and F are similar in form to Eq. (.), where the polarized PDFs and coe cients are replaced by the analogous unpolarized functions. Finally, the leading contribution to the g structure function is given by the Wandzura- Wilczek relation [6], g (x, Q )= g (x, Q )+ Z x dz z g (z,q )+O. (.3) Q

28 4 h p FIG..: Feynman diagram of semi-inclusive deep inelastic scattering. Similar to Fig.., except the struck quark fragments and an outgoing hadron h is tagged in the final state, in coincidence with the outgoing lepton. While g itself is not necessarily small, it is suppressed by factors of in the cross section asymmetries (Eq. (.9)). Consequently, the g structure function does not contribute in the Bjorken limit (Q!,finitex). In fact, A also vanishes in this limit, and the resulting A = g /F is a simple ratio of the polarized to unpolarized structure functions.. Semi-inclusive deep inelastic scattering In semi-inclusive DIS, the outgoing struck quark fragments and a single hadron is tagged in the final state (see Fig..). The polarized cross section asymmetries measured in SIDIS follow directly the discussion from DIS, apart from dependence on an additional kinematic variable z = p p h /p q, interpretedasthefractionofthetransferredvirtual photon momentum q being carried by the outgoing hadron with momentum p h. Experiments also typically measure cross sections that are dependent on the hadron s transverse momentum p h?. However, since our concern is only of the collinear distributions, the transverse momentum dependence is integrated out in the SIDIS observables presented in this section.

29 In the Bjorken limit, the SIDIS virtual photoproduction asymmetry is constructed as aratioofsemi-inclusivestructurefunctions, 5 A h (x, z, Q )= gh (x, z, Q ) F h (x, z, Q ), (.4) for a process in which a hadron h is identified in the final state. The semi-inclusive structure functions are subject to power suppressed corrections dependent on the outgoing hadron mass (M h /Q), in addition to the /Q and M/Q terms that arise in DIS. These so-called hadron mass corrections (HMCs) have recently been studied in Ref. [7]; however, since the Q values of the available experimental data are su ciently large, the /Q corrections in SIDIS are not considered in this work. The discussion here will again be restricted only to the leading contribution to the semi-inclusive structure functions. For the polarized g h function, it is given by g h (x, z, Q )= X q e q apple ( q(x, Q ) C qq (x, z) D h q (z,q )) + ( q(x, Q ) C gq (x, z) D h g (z,q )) (.5) + ( g(x, Q ) C qg (x, z) D h q (z,q )) + ( g(x, Q ) C gg (x, z) D h g (z,q )) + O. Q Identifying a hadron in the final state introduces a new nonperturbative function, Dq,g, h as a result of factorizing the hard scattering and hadronization distance scales. These are the fragmentation functions (FFs) and can be interpreted at LO as the probability for the struck quark to fragment into a jet containing a hadron h with fraction z of the virtual photon momentum. Furthermore, Eq. (.5) contains o -diagonal terms C qg and C gq related to the gluon occurring in the initial ( g) orfinal(d h g )state. Atnext-to-leading

30 e + h 6 e FIG..3: Feynman diagram of single-inclusive e + e -annihilation. The electron e and positron e + annihilate and a quark/anti-quark pair is produced from the intermediate photon or Z boson. The outgoing quarks fragment and a single hadron h is identified in the final state. order (NLO) in the s expansion, all except the diagonal glue-glue term ( C gg ), which enters at next-to-nlo (NNLO), contribute to the polarized structure function. As in the DIS case, the unpolarized structure function F h is defined similarly to g h with the polarized PDFs and hard scattering coe cients replaced by the unpolarized quantities. Semi-inclusive DIS plays a key role in global QCD analyses of spin dependent PDFs. Not only does it allow for separation of the quark and anti-quark flavors when combined with analysis of DIS observables, but will have sensitivity to sea quark helicity distributions that are favored in specific charged meson production. However, the determination of polarized PDFs from SIDIS is also strongly dependent on the parameterizations one chooses for the FFs, particularly for kaon production [8]. Therefore, it is important to discuss the QCD process from which much of our knowledge of FFs is obtained..3 Single-inclusive e + e annihilation Single-inclusive hadron production from e + e -annihilation (Fig..3) is perhaps the cleanest QCD process to study hadronization. Analogous to PDFs from DIS, SIA observables provide a direct probe to the nonperturbative FFs which describe the formation of

31 mesons and baryons from partons. The experimental observable for an outgoing hadron h is given by 7 F h (z,q )= tot d h dz (z,q ), (.6) where z =p h q/q is the fraction of the intermediate boson momentum q carried by the detected hadron with momentum p h,andq = p Q is the invariant mass. In the center-of-mass frame, the variable z =E h /Q can be interpreted instead as the energy fraction of the quark or anti-quark produced in the hard process that is carried by the outgoing hadron with energy E h. The di erential cross section in Eq. (.6) is normalized by the total inclusive e + e! q q cross section tot, which at NLO is tot(q )= X q 4 Q ẽ q + s(µ R ) + O( s). (.7) In this expression, = e /4 is the electromagnetic fine structure constant and ẽ q is defined as ẽ q = e q +e q g q V ge V (Q )+ g e A + g e V g q A + gq V (Q ). (.8) The coupling factor ẽ q is a sum of three terms related to quark-boson coupling. The first is the standard quark-photon coupling given by the quark electric charge e q.theadditional two terms dependent on and give the contributions from intermediate Z interference

32 8 and Z production, respectively, and are given by (Q ) = (Q ) = Q (MZ Q ) 4sin, (.9a) W cos W (MZ Q ) + MZ Z Q 4 4sin W cos, (.9b) W (MZ Q ) + MZ Z where M Z is the mass of the Z boson and Z is its width. The above expressions are also dependent on the weak mixing angle W,wheresin W /4. The quark vector and axial vector couplings in Eq. (.8) are g q V = 4 3 sin W and g q A =+,respectively,forthe up-type quark flavors (u, c). For the down-type quark flavors (d, s, b), they are given by g q V = + 3 sin W and g q A =.Lastly,theelectronvectorandaxialvectorcouplings are defined similarly by g e V = +sin W and g e A =. As in previous sections, the observable relevant here (Eq. (.6)) can be expressed in the collinear factorization framework as a convolution of the hard scattering coe cients H i with the parton-to-hadron FFs D h i, F h (z,q ) F h coll(z,q )= tot i X Hi Di h (z,q ), (.) where the sum over i runs over all parton flavors i = u, d, s,..., g. The FFs and hard coefficients are again dependent on a renormalization scale µ R that arises from the renormalization of final state divergences. As was done previously, we have set the renormalization scale µ R to the momentum transfer Q in Eq. (.). Many parallels can be drawn about the extraction of FFs from SIA with respect to PDFs from DIS. As in DIS, observables in SIA are sensitive only to the sum of quark and anti-quark flavor FFs, D h q + = D h q + D h q,andthereforerequireadditionalinputfrom SIDIS or other hadron production processes to separate the individual quark-to-hadron FFs. There is a distinction, however, in the treatment of heavy quark flavors, which are

33 9 generated perturbatively in the proton for DIS, but are prominent in SIA due to center-ofmass energies being much higher than the heavy quark production thresholds. Such topics will be left for discussion in Chapter 4..4 Observables in Mellin moment space Since nonperturbative functions are typically determined by experimental data, the theoretical observables discussed in the previous sections are implemented numerically in global QCD analyses. In the case where there is a significant amount of experimental data and many fit parameters, it becomes beneficial to improve the e ciency of the theoretical calculations. This can be achieved by computing observables in Mellin moment space, which is considerably faster than numerical calculations in x (or z) space[9]. Using the definition of the N-th Mellin moment of a function f(x), f(n) = Z dx x N f(x), (.) an observable O(x, Q ) defined by a single convolution integral in x space, e.g. Eqs. (.) or (.), can be expressed as a simple product in Mellin space, O(N,Q ) = X i H i (N,Q )f i (N,Q ). (.) Here the N-th moment of the observable, O(N,Q ), is given by general hard scattering coe cients H i and nonperturbative functions f i in Mellin space, summed over the different parton flavors i = u, d, s,..., g. Of course, in order to compare with experimental measurements, Eq. (.) must be reverted back to the x space expression O(x, Q ).

34 The Mellin inversion is performed using contour integration in complex moment space, O(x, Q ) = Z dnx N O(N,Q ), (.3) i C and is done numerically by defining N = c + ze i,wherecisthepoint of intersection of the contour with the real axis and is fixed to the right of the rightmost pole. The angle measures from the real axis to the contour line in complex space at c and is set to be 3 /4 toassurequickconvergenceoftheintegral. Eq.(.3)canthenbeexpressedasan integration over the variable z, O(x, Q ) = Z dzim e i x N O(N,Q ), (.4) where symmetry with respect to the real axis was applied. There are various approaches one could take to evaluate the integral over z, however,thestandardmethodiswitha Gaussian quadrature sum, O(x, Q ) ' X w i Im e i x N i O(N i,q ), (.5) i where w i is the Gaussian weight for the i-th point z i. This integration method is highly advantageous because it allows for a pre-computation of various theoretical quantities and thus significantly reduces computation time [9]. These are stored as Mellin tables, where the quantity of interest is evaluated at a collection of Mellin points (N i )definedbythe contour, which can subsequently be called upon in calculations of theoretical observables. The Mellin moment technique can also be applied to observables dependent on two variables x and x (e.g. x and z in Eq. (.5)). The double Mellin moment for a general observable O(x,x,Q )thatcontainstwononperturbativefunctionsf (x )andf (x )is

35 given by O(N,M,Q )= X i,j H i,j (N,M,Q )f,i (N,Q )f,j (M,Q ) (.6) where the N-th and M-th moment correspond to integrals over x and x,respectively. Returning to x space requires a double Mellin inversion, which is a straightforward extension of Eq. (.3) and is defined as O(x,x,Q )= Z ( i) dndmx N x M O(N,M,Q ). (.7) By defining N = c n + z n e i n and M = c m + z m e i m,andperformingthedoublemellin inversion as two subsequent single Mellin inversions, Eq. (.7) can be expressed as O(x,x,Q )= apple Z Re dz n Z dz m x N e i( n+ m) x M O(N,M,Q ) e i( n m) x M O(N,M,Q ). (.8) The double integration over z n and z m is then approximated as before by a Gaussian quadrature sum with Gaussian weights w i and w j, O(x,x,Q )= X X apple w i w j Re x N i i j n o e i( n+ m) x M j O(N i,m j,q ) e i( n m) x M j O(N i,mj,q ), (.9) where the summation occurs first over values of M j = c m + z j e i m followed by N i = c n + z i e i n. Typically the contours for N and M are set to be identical such that

36 c n = c m and n = m. For convenience, the hard coe cient functions needed to compute Eqs. (.), (.5), and (.) in Mellin space are given in Appendix A for the MS renormalization scheme [3, 3]. Besides the ability to pre-compute quantities in Mellin space, there are two additional advantages to using the Mellin moment technique in a global QCD analysis. The first is that integration over the kinematic variable x (or z) inmellinspacecanbecomputed analytically without spoiling the e ciency of the numerical calculation. This is especially relevant for experimental observables that require an averaging over kinematic bin ranges (excluding Q dependence). Consider, for example, an averaging between x min and x max for an observable O(x, Q ). Since the pre-factor x N in Eq. (.3) is the only x dependence, the averaged observable hoi can be written as O(x, Q ) Z = x bin (x max x min ) i C dn x max N x N min N O(N,Q ), (.3) and the contour integration can be performed using the Gaussian method discussed previously. Another key aspect of working in Mellin moment space is the application to scale evolution. In global QCD analyses, the nonperturbative distributions that enter into theoretical observables are parameterized at an input scale Q and must be evaluated at the scale of a given experiment. In the pqcd framework, the scale dependence is governed by the DGLAP equations [3 34], which in x space are given by df i (x, µ ) dln(µ ) = [P ij f j ](x, µ ), (.3) where f is a generic nonperturbative function that depends on a renormalization scale µ and P ij are the parton i! j splitting functions. Setting µ = Q then signifies the

37 dependence of the nonperturbative input on the momentum transfer of a QCD process. For PDF evolution, the variable x is the parton momentum fraction and the splitting functions are space-like (P S ij). When considering the scale dependence of FFs, x is replaced by the kinematic variable z and the splitting functions in Eq. (.3) become time-like (P T ij ). Solving the integro-di erential equations ((.3)) numerically in x space is highly 3 non-trivial [35 37]. Since convolution integrals become products in Mellin space, scale evolution becomes an ordinary coupled di erential equation, df i (N,µ ) dln(µ ) = P ij (N,µ )f j (N,µ ), (.3) which can be solved using the methods in Ref. [38]. Both the space-like and time-like splitting functions in Mellin space can be found in Appendix B. Clearly, the Mellin moment technique is an indispensable tool for time-intensive global QCD analyses, such as those that utilize rigorous Monte Carlo statistical methods. This will become even clearer in the following chapter, where multi-dimensional integrals from higher order /Q and nuclear corrections can be rendered as pre-computed quantities in Mellin space.

38 CHAPTER 3 Corrections Beyond the Parton Model In more recent years a wealth of high-precision DIS data at lower energies has become available from Je erson Lab [39 4], providing new insight into the spin structure of the nucleon. However, many of these new data exist at low values of Q and squared final state hadronic mass W,aphasespaceregiontypicallyexcludedbykinematiccutsinglobal QCD analyses. Furthermore, Je erson Lab provides fixed-target DIS data on nuclei such as deuterium [4, 43] and 3 He [44 46], which require proper treatment of bound nucleon e ects to extract quark spin information. In this chapter, we discuss finite-q and nuclear corrections relevant for experimental observables at Je erson Lab. 3. Higher Twist and Target Mass Corrections in DIS In the operator product expansion (OPE), a product of two operators at di erent space-time points can be written as a linear combination of local operators. This framework 4

39 allows one to construct hadronic matrix elements in DIS as a series expansion in twist 5 = d s, where the operators are defined by their mass dimension, d, minusspin,s. Higher twist (HT) operators from additional quark or gluon fields in the hard process generate terms that are suppressed by powers of (/Q) in DIS structure functions. Within the same formalism, there are target mass corrections (TMCs) of all orders in M/Q that originate from covariant derivative insertions in leading twist matrix elements [47]. Such corrections can be safely neglected if observables are evaluated in the Bjorken limit (Q!). On the other hand, when analyzing experimental data at values of momentum transfer on the order of the hadron mass Q M, corrections from HT operators and target mass become sizable and must be addressed. In the first of the three global QCD analyses presented in Chapter 5, where low-q DIS data from Je erson Lab are included, the HT corrections are treated up to twist-4 in the polarized structure function g and up to twist-3 for g, g = g ( ) + g ( 3) + g ( 4) + HT, (3.) g = g ( ) + g ( 3) + HT, (3.) where the leading twist g ( ) and g ( ) are given by Eqs. (.) and (.3), respectively, and HT denotes terms of higher twist. Furthermore, TMCs are included for the polarized structure functions up to twist-3 for both g and g.weshallseefromtheresultsofthe global analysis that these terms are su cient for describing low-q data since the twist-4 contribution is negligible for Je erson Lab kinematics, and the g structure function is suppressed by a factor /Q in the polarization asymmetries. Following the formalism of

40 6 Ref. [48], the target mass corrected g ( ) is g ( +TMC) (x, Q )= x Z 3 g( ) (,Q )+ ( ) apple dz (x + ) 4 z (3 ) ln z g ( ) (z,q ), (3.3) and the expression for the target mass corrected g ( ) is g ( +TMC) (x, Q )= x 3 g( ) (,Q )+ Z apple dz x 4 z ( ) + 3( ) ln z g ( ) (z,q ). (3.4) The TMCs are controlled by the Nachtmann variable = x/( + ) [47,49],where =+ and the Q scale dependence in the leading-twist structure functions is still given by the DGLAP equations, which can be derived in the OPE formalism from renormalized twist- operators. In the limit of large Q (! and! x), the polarized structure function g ( +TMC) returns to the massless limit expression g ( ) in Eq. (.), and the WW relation is recovered for g ( ) (Eq. (.3)). Another observation worth mentioning here is that the target mass corrected structure functions vanish at =andarethereforenonzerointhex! limit. This is known in the literature as the threshold problem [5 5], and has been a subject of interest in recent studies [53 56]. The issue, however, is relevant only for values of W that correspond to the nucleon resonance region, safely below the DIS region considered in global QCD analyses. Continuing now in the twist series for the polarized structure functions, the twist-3

41 7 part of g with TMCs is given by [48] g ( 3+TMC) (x, Q )= ( ) 3 D(,Q ) ( ) 4 Z dz z apple 3 (3 ) ln z D(z,Q ), (3.5) and for g it is g ( 3+TMC) (x, Q )= 3 D(,Q ) Z apple dz 3 + 3( ) ln z 4 z D(z,Q ). (3.6) Both are expressed in terms of the function D which is constructed as a sum over twist-3 parton distributions D q, D(x, Q )= X q e qd q (x, Q ). (3.7) Theoretically, these twist-3 parton distributions D q correspond to combinations of quarkgluon operators in the OPE and therefore provide information on nonperturbative quarkgluon interactions. Their scale dependence can be derived from renormalized twist-3 operators in the OPE and is highly nontrivial [57, 58]. We note here that extracting /Q power corrections from the general log(q ) behavior at NLO in the leading twist formalism is particularly challenging to accomplish with the existing data, and therefore distinguishing between di erent s approximations for the higher twist contributions is simply not feasible. For the purpose of this work, the evolution of the function D from the input scale Q is approximated by the large-n c limit, where N c corresponds to the number of active parton colors. Because the power corrections vary su ciently faster with Q than any modification to the large-n c limit expression for the higher twist evolution (from additional s corrections), the results would likely not be sensitive even if the large-n c s corrections were excluded altogether.

42 In the large-n c limit, the evolution of the twist-3 function is given in Mellin space as 8 D(N,Q ) s (Q e ) D(N,Q s (Q ), (3.8) ) where D(N,Q )isthemellinmomentofd(x, Q )ineq.(3.7). Theexponentoftheratio of strong couplings s is the anomalous dimension, e = ( N 3 f) (,N)+ E 4 +, (3.9) N where (,N)isthezerothorderpolygammafunctionand E is the Euler-Mascheroni constant. Since the twist-3 component of the polarized structure function g (Eq. (3.5)) contains an overall factor of, it clearly becomes zero in the limit of large Q. Interestingly, the twist-3 contribution to g (Eq. (3.6)) does not become zero in this limit, but instead reduces to g ( 3) (x, Q )=D(x, Q ) Z x dz z D(z,Q ), (3.) an expression similar to the Wandzura-Wilczek relation given by Eq. (.3) for the g ( ) term. Lastly, the twist-4 term of g is constructed at the hadron level, g ( 4) (x, Q )= H(x, Q ) Q, (3.) where H is a general function of fit parameters that will be determined by experimental data. In the QCD analysis studying the impact of Je erson Lab DIS data, the focus is primarily on determining the first two contributions in the twist series of the polarized structure functions. Consequently, both TMCs and Q evolution are not considered for

43 the twist-4 term, which is treated as background for the same reason that is given for the twist-3 evolution approximation Mellin moments and spin sum rules As mentioned in the previous chapter, calculating the target-mass corrected structure functions in Mellin space can significantly reduce computation time in a global QCD analysis, since the TMCs can be pre-computed and stored in Mellin tables. Consider, for example, the target-mass corrected twist- contribution to g. The expression given in Eq. (3.3) can be rewritten in the Mellin formalism as g ( +TMC) (x, Q )= i Z dn g ( +TMC) (N,Q ) x N+ + ( ) 3 4 Z dz z N+ apple (x + ) (3 ) ln z, (3.) where g +TMC (N,Q )isthemellinmomentofg +TMC (x, Q ) defined by Eq. (.). The quantity in brackets {...} is, in general, some function of the kinematics, M(x, N, Q ), that can be computed once beforehand for each value of N along the Mellin inversion contour. In fact, certain moments of the polarized structure functions themselves have significance in QCD and are worth mentioning here. The N =momentoftheg structure function is known as the Burkhardt-Cottingham (BC) sum rule [59], g (,Q )=, (3.3) which can be seen by taking the Mellin moment of the Wandzura-Wilczek relation (Eq. (.3)). Note that both target-mass corrected twist- and twist-3 contributions to the g structure function, given by Eqs. (3.4) and (3.6), satisfy the BC sum rule. Furthermore, there are